On Polynomial Time Local Decision

Authors Eden Aldema Tshuva , Rotem Oshman



PDF
Thumbnail PDF

File

LIPIcs.OPODIS.2023.27.pdf
  • Filesize: 0.73 MB
  • 17 pages

Document Identifiers

Author Details

Eden Aldema Tshuva
  • Tel Aviv University, Israel
Rotem Oshman
  • Tel Aviv University, Israel

Cite As Get BibTex

Eden Aldema Tshuva and Rotem Oshman. On Polynomial Time Local Decision. In 27th International Conference on Principles of Distributed Systems (OPODIS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 286, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.OPODIS.2023.27

Abstract

The field of distributed local decision studies the power of local network algorithms, where each network can see only its own local neighborhood, and must act based on this restricted information. Traditionally, the nodes of the network are assumed to have unbounded local computation power, and this makes the model incomparable with centralized notions of efficiency, namely, the classes 𝖯 and NP. In this work we seek to bridge this gap by studying local algorithms where the nodes are required to be computationally efficient: we introduce the classes PLD and NPLD of polynomial-time local decision and non-deterministic polynomial-time local decision, respectively, and compare them to the centralized complexity classes 𝖯 and NP, and to the distributed classes LD and NLD, which correspond to local deterministic and non-deterministic decision, respectively.
We show that for deterministic algorithms, requiring both computational and distributed efficiency is likely to be more restrictive than either requirement alone: if the nodes do not know the network size, then PLD ⊊ LD ∩ 𝖯 holds unconditionally; if the network size is known to all nodes, then the same separation holds under a widely believed complexity assumption (UP ∩ coUP ≠ 𝖯). However, when nondeterminism is introduced, this distinction vanishes, and NPLD = NLD ∩ NP. To complete the picture, we extend the classes PLD and NPLD into a hierarchy akin to the centralized polynomial hierarchy, and we characterize its connections to the centralized polynomial hierarchy and to the distributed local decision hierarchy of Balliu, D'Angelo, Fraigniaud, and Olivetti.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Local Decision
  • Polynomial-Time
  • LD
  • NLD

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Eden Aldema Tshuva and Rotem Oshman. Brief announcement: On polynomial-time local decision. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, pages 48-50, 2022. URL: https://doi.org/10.1145/3519270.3538463.
  2. Alkida Balliu, Gianlorenzo D'Angelo, Pierre Fraigniaud, and Dennis Olivetti. What can be verified locally? Journal of Computer and System Sciences, 97:106-120, 2018. URL: https://doi.org/10.1016/J.JCSS.2018.05.004.
  3. Michael R Fellows and Neal Koblitz. Self-witnessing polynomial-time complexity and prime factorization. Designs, Codes and Cryptography, 2(3):231-235, 1992. URL: https://doi.org/10.1007/BF00141967.
  4. Laurent Feuilloley and Pierre Fraigniaud. Randomized local network computing. In Proceedings of the 27th ACM symposium on Parallelism in Algorithms and Architectures, pages 340-349, 2015. URL: https://doi.org/10.1145/2755573.2755596.
  5. Pierre Fraigniaud, Mika Göös, Amos Korman, Merav Parter, and David Peleg. Randomized distributed decision. Distributed Computing, 27(6):419-434, 2014. URL: https://doi.org/10.1007/S00446-014-0211-X.
  6. Pierre Fraigniaud, Mika Göös, Amos Korman, and Jukka Suomela. What can be decided locally without identifiers? In Proceedings of the 2013 ACM symposium on Principles of distributed computing, pages 157-165, New York, NY, USA, 2013. ACM. URL: https://doi.org/10.1145/2484239.2484264.
  7. Pierre Fraigniaud, Magnús M Halldórsson, and Amos Korman. On the impact of identifiers on local decision. In International Conference On Principles Of Distributed Systems, pages 224-238, Berlin, Heidelberg, 2012. Springer. URL: https://doi.org/10.1007/978-3-642-35476-2_16.
  8. Pierre Fraigniaud, Juho Hirvonen, and Jukka Suomela. Node labels in local decision. Theoretical Computer Science, 751:61-73, 2018. URL: https://doi.org/10.1016/J.TCS.2017.01.011.
  9. Pierre Fraigniaud, Amos Korman, and David Peleg. Towards a complexity theory for local distributed computing. Journal of the ACM (JACM), 60(5):1-26, 2013. URL: https://doi.org/10.1145/2499228.
  10. William I Gasarch. Guest column: The third P=? NP poll. ACM SIGACT News, 50(1):38-59, 2019. URL: https://doi.org/10.1145/3319627.3319636.
  11. O. Goldreich and L. A. Levin. A hard-core predicate for all one-way functions. In Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing, STOC '89, pages 25-32, 1989. URL: https://doi.org/10.1145/73007.73010.
  12. Mika Göös and Jukka Suomela. Locally checkable proofs in distributed computing. Theory Comput., 12(1):1-33, 2016. URL: https://doi.org/10.4086/TOC.2016.V012A019.
  13. Joachim Grollmann and Alan L Selman. Complexity measures for public-key cryptosystems. SIAM Journal on Computing, 17(2):309-335, 1988. URL: https://doi.org/10.1137/0217018.
  14. Juris Hartmanis and Richard E Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285-306, 1965. Google Scholar
  15. Lane A Hemaspaandra and Jörg Rothe. Characterizing the existence of one-way permutations. Theoretical Computer Science, 244(1-2):257-261, 2000. URL: https://doi.org/10.1016/S0304-3975(00)00014-1.
  16. Christopher M Homan and Mayur Thakur. One-way permutations and self-witnessing languages. Journal of Computer and System Sciences, 67(3):608-622, 2003. URL: https://doi.org/10.1016/S0022-0000(03)00068-0.
  17. Marcin Jurdziński. Deciding the winner in parity games is in UP∩coUP. Information Processing Letters, 68(3):119-124, 1998. URL: https://doi.org/10.1016/S0020-0190(98)00150-1.
  18. Ker-I Ko. On some natural complete operators. Theoretical Computer Science, 37:1-30, 1985. URL: https://doi.org/10.1016/0304-3975(85)90085-4.
  19. Amos Korman and Shay Kutten. Distributed verification of minimum spanning trees. Distributed Computing, 20(4), 2007. URL: https://doi.org/10.1007/S00446-007-0025-1.
  20. Amos Korman, Shay Kutten, and Toshimitsu Masuzawa. Fast and compact self stabilizing verification, computation, and fault detection of an mst. In Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, pages 311-320, 2011. URL: https://doi.org/10.1145/1993806.1993866.
  21. Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. In Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing, pages 9-18, 2005. URL: https://doi.org/10.1145/1073814.1073817.
  22. Amos Korman, Shay Kutten, and David Peleg. Proof labeling schemes. In Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing, pages 9-18, 2005. URL: https://doi.org/10.1145/1073814.1073817.
  23. Moni Naor and Larry Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259-1277, 1995. URL: https://doi.org/10.1137/S0097539793254571.
  24. Fabian Reiter. A local perspective on the polynomial hierarchy. arXiv preprint, 2023. URL: https://arxiv.org/abs/2305.09538.
  25. Rachid Saad. Complexity of the forwarding index problem. SIAM Journal on Discrete Mathematics, 6(3):418-427, 1993. URL: https://doi.org/10.1137/0406033.
  26. Leslie G Valiant. Relative complexity of checking and evaluating. Information processing letters, 5(1):20-23, 1976. URL: https://doi.org/10.1016/0020-0190(76)90097-1.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail